Riemannian Metric Examples

Differential Geometry · Axiom Academy

EXAMPLE Computing with Riemannian Metrics Master calculations on curved manifolds through comprehensive examples Excellent work! You've mastered the fundamental computations with Riemannian metrics. Here's what we learned: Coordinate transformations: The metric tensor transforms as g' ij = (∂x k /∂x' i )(∂x l /∂x' j )g kl , allowing us to express the same geometry in different coordinates. Computing arc lengths: For a curve γ(t), the arc length is L = ∫√(g ij ẋ i ẋ j ) dt. This generalizes the Euclidean distance formula to curved spaces. Area and volume elements: The volume form is dV = √(det g) dx 1 ∧...∧dx n . The determinant encodes how the metric stretches or compresses space. Induced metrics: When embedding a surface in R n , the induced metric is g ij = ∂ i X · ∂ j X, obtained by pulling back the ambient metric. Metric properties: A Riemannian metric must be symmetric (g ij = g ji ) and positive definite (v T gv > 0 for all non-zero v), ensuring meaningful distances. Classical examples: The Euclidean, spherical, and hyperbolic metrics are the three constant curvature geometries in 2D, each with distinct properties and applications. Computational techniques: Master the use of the metric tensor to compute invariant geometric quantities: lengths, angles, areas, and curvature. These fundamental calculations form the foundation for studying geodesics, curvature, and the Einstein field equations in General Relativity!

This is the written version of the interactive lesson above. See the full Differential Geometry course.